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Permute the factors of a Kronecker product

0

$begingroup$

Let two matrices $A$ and $B$ of size $mtimes n$ and $p times q$, respectively.

What is the expression of two matrices $F$ and $G$ such that
$A otimes B = F ( B otimes A ) G$?

matrices kronecker-product

share|cite|improve this question

edited Apr 8 at 8:25

Rodrigo de Azevedo

13.2k41962

asked Apr 7 at 21:12

baptistebaptiste

53

$endgroup$

add a comment | 

0

$begingroup$

Let two matrices $A$ and $B$ of size $mtimes n$ and $p times q$, respectively.

What is the expression of two matrices $F$ and $G$ such that
$A otimes B = F ( B otimes A ) G$?

matrices kronecker-product

share|cite|improve this question

edited Apr 8 at 8:25

Rodrigo de Azevedo

13.2k41962

asked Apr 7 at 21:12

baptistebaptiste

53

$endgroup$

add a comment | 

$begingroup$

Let two matrices $A$ and $B$ of size $mtimes n$ and $p times q$, respectively.

What is the expression of two matrices $F$ and $G$ such that
$A otimes B = F ( B otimes A ) G$?

matrices kronecker-product

share|cite|improve this question

edited Apr 8 at 8:25

Rodrigo de Azevedo

13.2k41962

asked Apr 7 at 21:12

baptistebaptiste

53

$endgroup$

share|cite|improve this question

edited Apr 8 at 8:25

Rodrigo de Azevedo

13.2k41962

asked Apr 7 at 21:12

baptistebaptiste

53

share|cite|improve this question

edited Apr 8 at 8:25

Rodrigo de Azevedo

13.2k41962

asked Apr 7 at 21:12

baptistebaptiste

53

edited Apr 8 at 8:25

Rodrigo de Azevedo

13.2k41962

edited Apr 8 at 8:25

Rodrigo de Azevedo

13.2k41962

asked Apr 7 at 21:12

baptistebaptiste

53

asked Apr 7 at 21:12

baptistebaptiste

53

1 Answer
1

active

oldest

votes

0

$begingroup$

The matrices $F$ and $G$ are called commutation matrices. A commutation matrix $K_m,n$ is the unique permutation matrix that satisfies $K_m,n cdot rm vecX = rm vecX^T$ for any $X$ of size $m times n$, where $rm veccdot$ is the vectorization operator that stacks $X$ into a vector (column by column). These commutation matrices can be used to permute Kronecker products. They satisfy $$K_m,p^T cdot(A otimes B) cdot K_n,q = B otimes A,$$ where $A$ and $B$ are $m times n$ and $ptimes q$. Hence, your $F$ is $K_m,p$ and your $G$ is $K_n,q^T$. You can find a lot more details and properties about these matrices in [MN79, MN95].

[MN95] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, ZBL07044055, 1995.

[MN79] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.

share|cite|improve this answer

answered Apr 8 at 8:13

FlorianFlorian

1,5762721

$endgroup$

add a comment | 

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0

$begingroup$

The matrices $F$ and $G$ are called commutation matrices. A commutation matrix $K_m,n$ is the unique permutation matrix that satisfies $K_m,n cdot rm vecX = rm vecX^T$ for any $X$ of size $m times n$, where $rm veccdot$ is the vectorization operator that stacks $X$ into a vector (column by column). These commutation matrices can be used to permute Kronecker products. They satisfy $$K_m,p^T cdot(A otimes B) cdot K_n,q = B otimes A,$$ where $A$ and $B$ are $m times n$ and $ptimes q$. Hence, your $F$ is $K_m,p$ and your $G$ is $K_n,q^T$. You can find a lot more details and properties about these matrices in [MN79, MN95].

[MN95] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, ZBL07044055, 1995.

[MN79] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.

share|cite|improve this answer

answered Apr 8 at 8:13

FlorianFlorian

1,5762721

$endgroup$

add a comment | 

0

$begingroup$

The matrices $F$ and $G$ are called commutation matrices. A commutation matrix $K_m,n$ is the unique permutation matrix that satisfies $K_m,n cdot rm vecX = rm vecX^T$ for any $X$ of size $m times n$, where $rm veccdot$ is the vectorization operator that stacks $X$ into a vector (column by column). These commutation matrices can be used to permute Kronecker products. They satisfy $$K_m,p^T cdot(A otimes B) cdot K_n,q = B otimes A,$$ where $A$ and $B$ are $m times n$ and $ptimes q$. Hence, your $F$ is $K_m,p$ and your $G$ is $K_n,q^T$. You can find a lot more details and properties about these matrices in [MN79, MN95].

[MN95] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, ZBL07044055, 1995.

[MN79] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.

share|cite|improve this answer

answered Apr 8 at 8:13

FlorianFlorian

1,5762721

$endgroup$

add a comment | 

$begingroup$

The matrices $F$ and $G$ are called commutation matrices. A commutation matrix $K_m,n$ is the unique permutation matrix that satisfies $K_m,n cdot rm vecX = rm vecX^T$ for any $X$ of size $m times n$, where $rm veccdot$ is the vectorization operator that stacks $X$ into a vector (column by column). These commutation matrices can be used to permute Kronecker products. They satisfy $$K_m,p^T cdot(A otimes B) cdot K_n,q = B otimes A,$$ where $A$ and $B$ are $m times n$ and $ptimes q$. Hence, your $F$ is $K_m,p$ and your $G$ is $K_n,q^T$. You can find a lot more details and properties about these matrices in [MN79, MN95].

[MN95] Magnus, Jan R.; Neudecker, Heinz, Matrix differential calculus with applications in statistics and econometrics, ZBL07044055, 1995.

[MN79] Magnus, Jan R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Stat. 7, 381-394 (1979). ZBL0414.62040.

share|cite|improve this answer

answered Apr 8 at 8:13

FlorianFlorian

1,5762721

$endgroup$

share|cite|improve this answer

answered Apr 8 at 8:13

FlorianFlorian

1,5762721

answered Apr 8 at 8:13

FlorianFlorian

1,5762721

answered Apr 8 at 8:13

FlorianFlorian

1,5762721